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\begin{document}

\title{Dynamics of Graphene Stretching}

\author{M. Z. S. Flores$^{1}$}
 \email{zimmer@ifi.unicamp.br}
\author{G. Brunetto$^{1}$}
\author{P. A. S. Autreto$^{1}$}
\author{A. C. T. van Duin$^{2}$}
\author{D. S. Galv\~{a}o$^{1}$}
 \email{galvao@ifi.unicamp.br}
 
\affiliation{$^{1}$Instituto de F\'{i}sica ``Gleb Wataghin'', Universidade Estadual de Campinas, Unicamp, C.P. 6165, 13083-970, Campinas, S\~{a}o Paulo, Brazil}
\affiliation{$^{2}$Pennsylvania State University}

%\date{\today}% It is always \today, today,
             %  but any date may be explicitly specified
\date{\today}

\begin{abstract}

In the present work, we show reactive molecular dynamics study 
on the LAC formation from stretching graphene nanoribbons under different conditions, i.e. different temperatures, stretching velocities, widths and edges.
Our simulations show all the most important features observed experimentally and answers the question if wether a polyynic or cumulenic LACs
are expected to be formed under such experimental conditions.

\end{abstract}

\pacs{62.25.+g, 68.37.Lp, 68.66.La, 61.72.Nn}

%\keywords{Suggested keywords}

\maketitle

\section{Introduction}

Carbon is the main compound of organic molecules, playing a major role in biomolecules, such as proteins and DNAs. 
Such overspread of carbon-based compounds in nature
can be easily addressed to its peculiar eletronic properties, resulting from three distinct orbital hybridization 
(namely sp$^3$, sp$^2$ and sp), which grant its presence on a myriad of distinct electronic environment. 
However, carbon allotropes have been more commonly found to occur in 
either sp$^3$ (diamond) or sp$^2$ (graphene, nanotubes and fullerenes) hybridizations. Although $sp$-based carbon allotropes have long been 
hypothesized \cite{Hoffman1966} under the form of linear atomic chains (LAC), only in the last two decades the 
synthesis of its candidates has been rationally achieved \cite{Rinzler1995,Lagow1995,Roth1996,Wang2000,Zhao2003,Bowling2006,Jin2009,Chuvilin2009,Borrnert2010}. 
Contrary of what is found for metallic chains, which are expected to appear as chain of only a few atoms \cite{Galvao,Ugarte}, it has been shown that its carbon
counterpart may have hundreds of atoms when formed inside or attached to nanotubes \cite{Rinzler1995,Wang2000,Zhao2003}. Despite evidences
showing the possible existence of such big LAC, its isolation and/or its solid state polymerization has been a major challenge
to experimentalists \cite{Baughman2006}, and so far only somewhat small carbon LACs (tens of C atoms) have been observed under the 
aforementioned conditions \cite{Lagow1995,Roth1996,Bowling2006,Jin2009,Chuvilin2009}, being the biggest ones found in solution \cite{Lagow1995,Roth1996,Bowling2006}.

Because of the distinct nature of the $\pi$ bonds present in carbon LAC, it has been subject of inspiring discussions if whether such chains occur as
polyynes ($\cdots\rm{C}\equiv\rm{C}-\rm{C}\equiv\rm{C}\cdots$) or polycumulene ($\cdots\rm{C}=\rm{C}=\rm{C}=\rm{C}\cdots$) \cite{Yang2008,Fan2009,Hu2009},
which is still an open question, but it's quite widely accepted that this behavior is highly dependent on chain parity (odd number of carbon
in a LAC are polyynic, while even are cumulenic). Recently, Jin \emph{et. al} \cite{Jin2009} and Chuvilin \emph{et. al.}\cite{Chuvilin2009} have 
independently performed HRTEM experiments similar to those previously carried out to metals \cite{Galvao,Ugarte}, where a high current beam 
is appllied on suspended sheets of the material of interest. As result, such current creates a neck in the graphene sheet because of  
two near holes are formed by the such current. They observed the time evolution of such neck and its subsequent LAC formation.
Such experiments raised questions on the dynamic of the LAC formation and rupture as key information on assembly of nanocontacts. 
Wang \emph{et. al.} \cite{Wang2007,Wang2009} performed room temperature reactive potential (Brenner-type) molecular dynamics simulations and showed the formation
of huge carbon LACs (over 100 atoms) as result of unraveling carbon from defective graphene sheets under stretch. 
They have also performed Car-Parrinello \emph{ab initio} MD simulations and showed that such unraveling might occur, which would be responsible
for LAC formation. However, it is known that Brenner potential in its original form leads to a serious overestimation of critial loads and
shear stresses in fracture mechanics, due to a fast energy vanishing in its formulation. In order to overcome this, Pastewka \textit{et. al.}
have proposed a screening function in the Brenner potential, avoiding unphysical results \cite{Pastewka2008}. 
The carbon chemistry related to triple bonds remains a challenge in Brenner-like potentials, though. 

More recently, Hobi Jr. \emph{et. al}\cite{Hobi2009}, have also used room temperature \emph{ab initio} MD simulations and showed the LAC formation, 
however they observed its rupture after only a few atoms were present in LAC. This way, it is of our understanding that although these work
does elucidate important features, they give only limited access to the experimental features
observed by Jin \emph{et. al} \cite{Jin2009} and Chuvilin \emph{et. al.}\cite{Chuvilin2009}. In the present work, we show reactive molecular dynamics study 
on the LAC formation from stretching graphene nanoribbons under different conditions, i.e. different temperatures, stretching velocities, widths and edges.
Our simulations show all the most important features observed experimentally and answers the question if wether a polyynic or cumulenic LACs
are expected to be formed under such experimental conditions.


%\begin{figure}[b]
%\begin{center}
%\includegraphics[scale=0.4]{fig01-numbered.eps}
%\end{center}
%\caption{Buckycatcher structure.}
%\label{fig01}
%\end{figure}


\section{Methodology}

We have performed molecular dynamics simulation for graphene nanoribbons (GNR) of different sizes under periodic boundary condition in one direction 
(as indicated by arrows in Fig. \ref{fig:scheme}). Initially the single-layer graphene ribbons were cut to form the zig-zag (ZZ) and 
arm-chair (AC) pattern related to the ribbon's edge with three different sizes taken as input (Figure \ref{fig:scheme}). 
Then, the cell size was slowly increased under constant stretching velocities of different magnitudes. The GNR sizes were addressed
as the number WxL, such as W and L are the number of carbon atoms perpedicular (as show in Fig. \ref{fig:scheme} as black dots) and
parallel (as show in Fig. \ref{fig:scheme} as red dots) to the periodic direction, respectivelly. The values of W were 4, 6 and 8 regardless the type
of edge, while L was taken as 10 and 16 for zigzag and armchair, respectively.

In this work, the two possible edges (zigzag and armchair) were subject to four different continuous stretching velocities, namely 
$1.0 \times 10^{-4}$, $2.0 \times 10^{-4}$, $4.0 \times 10^{-4}$ and $100 \times 10^{-4}$\thinspace \AA/fs. Different hydrogen
concetrations were used to passivate GNR borders, namely 0\%, 20\%, 40\%, 60\% and 100\%.
Initially, all molecules present in this work were submitted 
to structure optimization within the conjugate gradient before stretching. Afterwards, NVT molecular dynamics simulations with
velocity verlet integrator and berendsen thermostat were employed
with a timestep of $0.5$\thinspace fs (saving the initial $1.0$\thinspace ps for thermal equilibration). Four 
temperatures (100K, 300K, 600K and 900K) were used in our simulations for all H concentrations in GNRs.
The simulation lasted until full rupture of GNRs were observed and no LACs could possibly be formed as result of
the applied stretch. This methodology is similar to the one used on previous work of metallic nanowires \cite{Galvao}.

All simulations were carried out using the binding energy bond-order (BEBO) approach as developed by A. C. T. van Duin \emph{et. al.} and implemented in 
ReaxFF \cite{reaxff2001,reaxff2008}, which is a software for classical molecular mechanics (MM) and molecular 
dynamics (MD) simulations suitable for reactive systems, i.e. bonds are allowed to brake and/or form and the atomic hybridizations 
are allowed to change during MM/MD simulations as well.

\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figuras/scheme.eps}
\caption{(color online) Schematic representation of the edge termination for the GNR used in our calculations. (a) zig-zag and (b) arm-chair.}
\label{fig:scheme}
\end{figure}


%According to Iijima \cite{Jin2009}, the current used to put holes into the graphene sheet was big enough to 
%lead to the well-known knock-on effect, thus no hydrogen is expected to be present in the structure. 
%However, for the interest of completeness we have used the same aforementioned inputs with hydrogen atoms bound to the 
%carbons of the ribbon's edge to perform the same simulations for one particular stretching velocity. 
%Also non-symmetric ribbon cuts were simulated, because of the holes made by the STM tip doesn't 
%account for controlling its size and shape. For that, we have taken ZZ edge ribbon and cut it diagonally along both 
%ZZ and AC patterns. The same was made for the AC edge ribbons. For these situations, 
%again only one stretching velocity was applied, but both perpendicular in-plane directions were used in the calculations.
%
%
%All calculations were performed under the binding energy bond-order (BEBO) framework as developed by A. C. T. van Duin \emph{et. al.} and implemented in ReaxFF \cite{reaxff2001,reaxff2003,reaxff2008}, which is a software for classical molecular mechanics (MM) and molecular dynamics (MD) simulations suitable for reactive systems, i.e. bonds are allowed to brake and/or form and the atomic hybridizations are allowed to change during MM/MD simulations as well.
%


ReaxFF performs calculations for periodic structure using cartesian coordinates for the in-cell atoms, 
thus any change in lattice doesn't produce any instantaneous atomic rearrangement, as it would if internal coordinates 
were used instead. This way, the only consequence of increasing the lattice each step occurs to the bond that binds 
atoms from neighbor cells, thus the lattice increase at each step has to be small enough to allow the system 
to rearrange itself in order to counterbalance such external stress, but not too small so the net change in lattice size is irrelevant.

\section{Results and Discussion}

Our simulations showed LAC formation for most of the cases studied, regardless of initial condition. However, it could be
noticed a clear preference on LAC formation when a zigzag edge is in place, which can be explained by the lower
energy barrier for bond bending when compared to bond stretching. This way, almost 100\% of the zigzag edge cases studied
resulted in LAC, while this number decreased to 89\% when armchair edges were in place.
 In average, our simulations showed LAC formed by seven atoms per chain, but we could observe particular cases where 
a LAC was formed by 15 carbon atoms, which is in good agreement with experiment \cite{Jin2009,Chuvilin2009}. 

%%
%%
%%

\begin{figure}[b]
\centering
\includegraphics[width=\columnwidth]{Figuras/lacs.eps}
\caption{Characteristic formation of CLAC during MD simulations. (a) single LAC diagonaly oriented; (b) fibrous regime; (c) long single stranded CLAC; (d) fibrous regime with two carbon rings in the middle; (e) fibrous regime with curved LAC and (f) fibrous regime with one carbon ring in the middle.}
\label{fig:lacs}
\end{figure}

Different carbon linear atomic chains (CLAC) are shown in Figure \ref{fig:lacs} at graphene nanoribbons 
under stretch. It is worth noting that many of the structures present here resemble the structures observed by 
Jin \emph{et. al.} \cite{Jin2009}. We have observed that the double strand tend to appear on the 
armchair nanoribbons, while the greater one strand CLAC was observed on the zig-zag nanoribbons. 
Further analysis on the dynamical process of the CLAC formation show that the atoms from the edge play 
an important role in the CLAC formation. Such behavior is similar to the one found in metals \cite{Galvao}. 


Although the hydrogen influence is not accessible under the experimental point of view, we have performed simulations on
graphene nanoribbons with borders passivated by the presence of hydrogen in different concentrations, ranging from zero
(all carbon atoms from the edge had dangling bonds) to 100\% (no dangling bonds were initially present in the structure). 
In a simple count, the hydrogen presence affected little
on the LAC existence, since most of our simulations resulted in chain formation. However, the higher the H concentration the
lower the occurence of chain formation. As a general remark, the H presence affected mainly on LAC lifetime, since
it was observed that the chain with hydrogen incorporated tend to weaken the carbon-carbon bond, thus leading to shorter
and easier to break chains than those without H. This way, while almost 100\% of cases resulted in LAC formation for the two
lowest H concentration, only 80\% of the simulations for totally passivated ribbon edge resulted in chain occurence.


Figure \ref{fig:snapshot} shows the time evolution snapshots for the stretching proccess for the graphene nanoribbon 
shown in Fig \ref{fig:lacs}c with cutoff distance of $1.8 \AA$. The color scheme indicates the relative difference 
($\Delta L_b$) in the neighbouring carbon distances taking 
as reference the initial average bond length (i.e. $L_0=1.402$\AA). One can observe that initially the bond length distribution 
remains almost constant (less than $3\%$) with respect to $L_0$. As stretching starts taking place, one can find a broader
distribution on bond lengths. It is worth noting that although some vertical (in stretching direction) bonds increase, there
are horizontal (perpendicular to stretching) bonds in the middle of the nanoribbon that remain unchanged, which forces 
it to maintain its original structure.
This way, when rupture takes place there will be parts of the graphene nanoribbon with shorter bond lenghts, thus higher
energy for its dissociation is needed, and one can observe the formation of several atomic chains forming a fibrous state (as shown at
instants 22.75ps and 30.00ps). This intermediate structure is in full agreement with experimental observations \cite{Chuvilin2009}. 


\begin{figure}
\centering
\includegraphics[width=\columnwidth]{Figuras/snapshots.eps}
\caption{(color online) Snapshots for the time evolution of graphene under stretch for the case shown in Fig. \ref{fig:lacs}c (H atoms not shown for sake of clarity). $\Delta L_b$ indicates the relative difference on the instant bond length with respect to the initial average bond length.}
\label{fig:snapshot}
\end{figure}


By the time stretching continues to act upon the GNR such fibrous regime begins to disappear and less atomic chains become apparent 
($t=34.50ps$), until a single atomic chain is present. By analysing the CLAC, we found that most of our simulations lead to 
polyynic-like ($\cdots\rm{C}\equiv\rm{C}-\rm{C}\equiv\rm{C}\cdots$) structures rather than
polycumulenic-like ($\cdots\rm{C}=\rm{C}=\rm{C}=\rm{C}\cdots$). It is in accordance to  Jin \emph{et. al.} results, where
polycumulene is mainly observed when some geometrical symmetry is found in the suspended atomic chain, thus making its
observation rather difficult. This way, we could observe the polycumulenic-like structure formation only when high symmetric
rupture was present ($t=45.85ps$ and $t=47.75ps$), which is quite rare to happen and difficult to access experimentally. 
Since bond dissociation requires lower energy
to happen when high bond length alternation is present (i.e. polyyne), the cumulenic structure still evolve by breaking carbon
bond of a ring in the LAC-GNR junction until polyynic-like structure is present ($t=50.70ps$), which is the final stage before rupture.
It is worth noting that the overall graphene nanoribbon tearing process occurs as result of localizing strain, such that
fibrous state and LAC formation forces the strain localization in this relevant region, while the remaining parts of GNR
structure manages to keep a bond length distribution similar to the initial unstretched structure.


Acknowledgments - This work was supported in part
by the Brazilian Agencies CNPq, CAPES and FAPESP.

this is a line test.
I would like to check some git features.
try the fetch


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\end{document}
